Almost Global Existence for the 3D Prandtl Boundary Layer Equations

In this paper, we prove the almost global existence of classical solutions to the 3D Prandtl system with the initial data which lie within ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon $\end{document} of a stable shear flow. Using anisotropic Littlewood-Paley energy estimates in tangentially analytic norms and introducing new linearly-good unknowns, we prove that the 3D Prandtl system has a unique solution with the lifespan of which is greater than exp(ε−1/log(ε−1))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\exp (\varepsilon ^{-1}/\log (\varepsilon ^{-1}))$\end{document}. This result extends the work obtained by Ignatova and Vicol (Arch. Ration. Mech. Anal. 2:809–848, 2016) on the 2D Prandtl equations to the three-dimensional setting.